Problem 98 - A Graphical Approach to Precalculus with Limits

Question

Starting at point \(X,\) a ship sails 15.5 kilometers on a bearing of \(200^{\circ}\), then tums and sails 2.4 kilometers on a bearing of \(320^{\circ} .\) Find the distance of the ship from point \(X .\)

Step-by-Step Solution

Answer

The ship is 14.46 km from point X.

Step 1: Draw the Problem

Before solving, visualize the problem by drawing a diagram. Start with point \(X\), plot the first path of 15.5 km on a bearing of \(200^{\circ}\), then plot the second path of 2.4 km on a bearing of \(320^{\circ}\). The two paths form parts of two different angles from the North direction. Remember that the bearing is measured clockwise from the north.

Step 2: Determine Angles Between Paths

The first path is on a \(200^{\circ}\) bearing, which means it forms a \(20^{\circ}\) angle with the south direction. Similarly, the second path on a \(320^{\circ}\) bearing forms a \(40^{\circ}\) angle with the north direction. Based on the bearings, the angle between the two paths within the triangle is the exterior angle, thus \(60^{\circ}\).

Step 3: Use the Law of Cosines

To find the distance from the final position back to point \(X\), use the Law of Cosines in the triangle formed. Let that distance be \(d\): \[d^2 = 15.5^2 + 2.4^2 - 2 \times 15.5 \times 2.4 \times \cos(60^{\circ})\]

Step 4: Calculate the Law of Cosines

Substitute the values and solve:\[d^2 = 15.5^2 + 2.4^2 - 2 \times 15.5 \times 2.4 \times 0.5\]Calculate:\[15.5^2 = 240.25\]\[2.4^2 = 5.76\]\[2 \times 15.5 \times 2.4 \times 0.5 = 37.2\]\[d^2 = 240.25 + 5.76 - 37.2\]\[d^2 = 208.81\]

Step 5: Solve for the Distance

Calculate the square root of \(d^2\):\[d = \sqrt{208.81} = 14.46\]The ship is approximately 14.46 kilometers from point \(X\).

Key Concepts

Law of Cosines

The Law of Cosines is a key tool in solving triangles when you don’t have a right angle, which makes it extremely helpful in navigation problems. Just like the Pythagorean theorem is used in right triangles, the Law of Cosines extends to any type of triangle. The formula is:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Where:

• \( c \) is the side opposite angle \( C \)
• \( a \) and \( b \) are the other two sides
• \( C \) is the angle you're using in the calculation

The primary purpose of the Law of Cosines is to find a missing side of a triangle when you’re given two sides and the angle between them, or to find an angle when all three sides are known. In the ship's journey problem, it’s used to find the ship's distance from point \( X \) after two directional movements. Knowing how to apply this law is crucial, especially when the angles aren't easy to visualize as they aren’t right angles.

Bearings in Navigation

Bearings are a common way to indicate direction in navigation. They are measured in degrees from the North, moving clockwise. Think of them as a means of setting your compass. In the problem, the ship uses bearings of \(200^{\circ}\) and \(320^{\circ}\) to navigate.
To visualize, imagine a compass. A bearing of \(0^{\circ}\) points directly north, and \(90^{\circ}\) points east. Therefore, a bearing of \(200^{\circ}\) means the ship moves south and slightly to the west. It forms a \(20^{\circ}\) angle with the south direction. Similarly, a bearing of \(320^{\circ}\) points north and slightly to the west, forming a \(40^{\circ}\) angle with the north direction.
This understanding helps when determining the angle between different paths. For instance, summing the angles away from the north gives us the internal angle within the formed triangle, essential for the Law of Cosines.

Geometric Visualization

Visualization plays a vital role in solving geometry problems. By drawing the paths and angles, each step becomes clearer.
In this exercise, visualizing involves two main actions:

• Sketching the two paths the ship takes from point \( X \).
• Considering the angles, which entails realizing how these bearings relate to each other on the compass.

Drawing a diagram helps in connecting the abstract angles and side lengths with tangible paths.
Begin by plotting the first path on the given bearing of \(200^{\circ}\) and the second on \(320^{\circ}\). Next, visualize or measure the angles they create with the north-south line.
With these visuals, calculating the angle between the two paths becomes simpler, revealing that the needed angle for the Law of Cosines is the \(60^{\circ}\) between the tracks of travel. This geometrical insight transforms numerical data into a mental map, making the path clear.